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Checkerboard-free topology optimization using polygonal finite elements
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Motivation
• In topology optimization, parameterization of shape and topology of the
design has been traditionally carried out on uniform grids;
• Conventional computational approaches use uniform meshes consisting of
Lagrangian-type finite elements (e.g. linear quads) to simplify domain
discretization and the analysis routine;
• However, as a result of these choices, several numerical artifacts such as
the well-known “checkerboard” pathology and one-node connections may
appear;
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Motivation
• In topology optimization, parameterization of shape and topology of the
design has been traditionally carried out on uniform grids;
• Conventional computational approaches use uniform meshes consisting of
Lagrangian-type finite elements (e.g. linear quads) to simplify domain
discretization and the analysis routine;
Checkerboard:
One-node hinges:
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Motivation
• In this work, we examine the use of polygonal meshes consisting of convex
polygons in topology optimization to address the aforementioned issues
P
2.792
2.2
1.7921.0
0.788
T. Lewinski and G. I. N. Rozvany. Exact analytical solutions for some popular benchmark problems in
topology optimization III: L-shaped domains. Struct Multidisc Optim (2008) 35:165–174
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Motivation
• In this work, we examine the use of polygonal meshes consisting of convex
polygons in topology optimization to address the abovementioned issues
T. Lewinski and G. I. N. Rozvany. Exact analytical solutions for some popular benchmark problems in
topology optimization III: L-shaped domains. Struct Multidisc Optim (2008) 35:165–174
Solution obtained with 9101 elements
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Outline
• Polygonal Finite Element
• Topology optimization formulation
• Numerical Results
• Concluding remarks
• Ongoing work
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Polygonal Finite Element
• Isoparametric finite element formulation constructed using Laplace shape
function.
Pentagon Hexagon Heptagon
• The reference elements are regular n-gons inscribed by the unit circle.
N. Sukumar and E. A. Malsch. Recent advances in the construction of polygonal finite element interpolants.
2006. Archives of Computational Methods in Engineering, 13(1):129--163
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Polygonal Finite Element
• Isoparametric finite element formulation constructed using Laplace shape
function.
• Isoparametric mapping
N. Sukumar and E. A. Malsch. Recent advances in the construction of polygonal finite element interpolants.
2006. Archives of Computational Methods in Engineering, 13(1):129--163
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Polygonal Finite Element
• Laplace shape function
Non-negative
Linear completeness
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Polygonal Finite Element
• Laplace shape function for regular polygons
• Closed-form expressions can be obtained by employing a symbolic
program such as Maple.
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Polygonal Finite Element
• Numerical Integration
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Outline
• Polygonal Finite Element
• Topology optimization formulation
• Numerical Results
• Concluding remarks
• Ongoing work
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Topology optimization formulation
• The discrete form of the problem is mathematically given by:
• minimum compliance
• compliant mechanism
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Relaxation
• The Solid Isotropic Material with Penalization (SIMP) assumes the following
power law relationship:
• In compliance minimization, the intermediate densities have little stiffness compared to their contribution to volume for large values of p
Sigmund, Bendsoe (1999)
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Outline
• Polygonal Finite Element
• Topology optimization formulation
• Numerical Results
• Concluding remarks
• Ongoing work
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Numerical Results (Compliance Minimization)
• Cantilever beam
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Cantilever Beam Compliance Minimization
(a) (b)
(c) (d)
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Numerical Results (Compliant Mechanism)
• Force inverter
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Force Inverter Compliant Mechanism
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Higher Order Finite Element
• Michell cantilever problem with circular support
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Higher Order Finite Element
Solution based on a Voronoi meshSolution based on a T6 mesh
• Michell cantilever problem with circular support
Talischi C., Paulino G.H., Pereira A., and Menezes I.F.M. Polygonal finite elements for topology optimization: A
unifying paradigm. International Journal for Numerical Methods in Engineering, 82(6):671–698, 2010
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Outline
• Polygonal Finite Element
• Topology optimization formulation
• Numerical Results
• Concluding remarks
• Ongoing work
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Concluding remarks
• Solutions of discrete topology optimization problems may suffer from
numerical instabilities depending on the choice of finite element
approximation;
• These solutions may also include a form of mesh-dependency that
stems from the geometric features of the spatial discretization;
• Unstructured polygonal meshes enjoy higher levels of directional
isotropy and are less susceptible to numerical artifacts.
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Ongoing research
• Well-posed formulation of topology optimization problem based on
level set (implicit function) description and extension to other
objective functions.
P=1
80
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